Mathematical morphology
- with emphasis on analysis of hyperspectral
images and remote sensing applications
Yuliya Tarabalka12, Jón Atli Benediktsson1,
Jocelyn Chanussot2
1 University of Iceland,
2 Grenoble Institute of Technology, France
([email protected])
Outline
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Introduction
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Basic concepts of mathematical morphology
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Mathematical morphology for grey-scale and hyperspectral
images
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Remote sensing application 1: Classification of hyperspectral
images of an urban area using morphological profiles
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Remote sensing application 2: Segmentation and classification
of hyperspectral images using watershed
•
Practical session on mathematical morphology
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Basic concepts of
mathematical morphology
Mathematical morphology: why to use?
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How to remove this noise?
How to separate these two components?
How to label differently these two connected
shapes?
How to compare these two shapes?
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Mathematical morphology (MM)
• A theory for the analysis of spatial structures
• Morphology: aims at analysing the shape and form of objects.
• Mathematical: analysis is based on:
¾ set theory
¾ integral geometry
¾ lattice algebra
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Non-linear processing operators (do not blur the edges as
convolutions do)
We’ll concentrate on MM for digital images: binary, grey-scale
and hyperspectral
Basic idea: locally compare structures within the image with a
reference shape called the Structuring Element (SE)
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Structuring element (SE)
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A small set used to analyse locally the image
Shape and size of SE Å a priori knowledge about the
geometry of relevant/irrelevant image structures
Usually symmetrical, connected, and convex
But not always
origins of SEs
for positioning of
the SE at a given pixel
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Dilation and erosion
Fundamental morphological operators =
2 letters of the morphological alphabet
All other operators are expressed in terms of
dilations and erosions
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Dilation for binary images
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“Does the SE hit the set?”
Dilation of a set X by a structuring element E is defined as
the locus of points x such that E hits X when its origin is
placed at x:
δE(X) = {x ∈ Rd | Ex ∩ X ≠ ∅}
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result of dilation
E
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Dilation for binary images
•
•
“Does the SE hit the set?”
Dilation of a set X by a structuring element E is defined as
the locus of points x such that E hits X when its origin is
placed at x:
δE(X) = {x ∈ Rd | Ex ∩ X ≠ ∅}
E
result of dilation
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Dilation: properties and use
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Basic property: X ⊆ δE(X)
Consequences:
¾ Fill in the holes smaller than E
¾ Enlarge capes
¾ Connect two close shapes
Example of application: bridging gaps
E
R. C. Gonzalez, R. E. Woods, Digital Image Processing. Prentice Hall, 2002.
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Erosion for binary images
•
•
“Does the SE fit the set?”
Erosion of a set X by a structuring element E is defined as the
locus of points x such that E is included in X when its origin
coincides with x:
εE(X) = {x ∈ Rd | Ex ⊆ X }
E
erosion?
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Erosion for binary images
•
•
“Does the SE fit the set?”
Erosion of a set X by a structuring element E is defined as the
locus of points x such that E is included in X when its origin
coincides with x:
εE(X) = {x ∈ Rd | Ex ⊆ X }
E
erosion
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
!
Mathematical morphology
Erosion for binary images
•
•
“Does the SE fit the set?”
Erosion of a set X by a structuring element E is defined as the
locus of points x such that E is included in X when its origin
coincides with x:
εE(X) = {x ∈ Rd | Ex ⊆ X }
E
erosion
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Erosion for binary images
•
•
“Does the SE fit the set?”
Erosion of a set X by a structuring element E is defined as the
locus of points x such that E is included in X when its origin
coincides with x:
εE(X) = {x ∈ Rd | Ex ⊆ X }
E
result of erosion
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Erosion: properties and use
•
•
•
Basic property: εE(X) ⊆ X ⊆ δE(X)
Consequences:
¾ Eliminate connected components smaller than E
¾ Eliminate narrow capes
¾ Enlarge holes
Example of application: eliminating irrelevant details (noise)
erosion
with 11x11
square SE
squares 10x10
noise is
removed
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
large shapes
are shrinked
Mathematical morphology
Erosion: properties and use
•
•
•
Basic property: εE(X) ⊆ X ⊆ δE(X)
Consequences:
¾ Eliminate connected components smaller than E
¾ Eliminate narrow capes
¾ Enlarge holes
Example of application: eliminating irrelevant details (noise)
squares 10x10
erosion
dilation
with 11x11
square SE
with 11x11
square SE
noise is
removed
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
large shapes
are shrinked
sizes of large
objects are
restored!
Mathematical morphology
Opening for binary images
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Opening of a set X by a structuring element E is defined as
the erosion of X by E followed by the dilation with the reflected
(symmetric with respect to the origin) SE Ẽ:
γE(X) = δẼ[εE(X)]
•
Consequences:
¾ Objects smaller than E disappear
¾ Other objects remain “unchanged”
erosion
dilation
with 11x11
square SE
with 11x11
square SE
squares 10x10
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
result of opening
Mathematical morphology
Opening for binary images
•
Opening of a set X by a structuring element E is defined as
the erosion of X by E followed by the dilation with the reflected
(symmetric with respect to the origin) SE Ẽ:
E
γE(X) = δẼ[εE(X)]
•
Consequences:
¾ Objects smaller than E disappear
¾ Other objects remain “unchanged”
Ẽ
erosion
dilation
with 11x11
square SE
with 11x11
square SE
squares 10x10
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
result of opening
Mathematical morphology
Opening for binary images
•
Opening of a set X by a structuring element E is defined as
the erosion of X by E followed by the dilation with the reflected
SE Ẽ:
γE(X) = δẼ[εE(X)]
•
Consequences:
¾ Objects smaller than E disappear
¾ Other objects remain “unchanged”
E
result of opening
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology
Closing for binary images
•
Closing of a set X by a structuring element E is defined as the
dilation of X by E followed by the erosion with the reflected SE Ẽ:
φE(X) = εẼ[δE(X)]
• Order: γE(X) ⊆ X ⊆ φE(X)
•
Consequences:
¾ Holes smaller than E are eliminated
¾ Other objects remain “unchanged”
E
E
result of dilation
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
result of closing
Mathematical morphology
Other MM operators: Top-hat
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Top-hat transformation of an image X is defined as the
difference between the original image X and its opening γ:
TH(X) = X - γ (X)
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Consequences:
¾ Objects smaller than E are extracted
¾ Other objects disappear
_
X
=
γ11x11(X)
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
TH(X)
Mathematical morphology
Morphological gradient
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The basic morphological gradient of an image X is defined as
the arithmetic difference between the dilation and the erosion of
X by the elementary SE E :
ρ E(X) = δE(X) - εE(X)
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Only symmetric SEs are considered
Tends to depend less on edge directionality (when compared to
first derivatives)
E
X
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
morphological
gradient of X
Mathematical morphology
Summary: Basic concepts of MM
•
Mathematical morphology can be defined as a theory for the
analysis of spatial structures
•
Basic idea of MM: locally compare structures within the image
with a reference shape called the Structuring Element
•
Dilation and erosion are two fundamental MM operators
Î All other operators are expressed in terms of dilations and
erosions
•
Dilation, erosion, opening, closing, top-hat and
combinations of these operators are often used
for image filtering
Y. Tarabalka, J. A. Benediktsson, J. Chanussot
Mathematical morphology